A216316 G.f.: 1/( (1-8*x)*(1+x)^2 )^(1/3).

1, 2, 13, 80, 538, 3740, 26650, 193160, 1417945, 10511450, 78533629, 590485208, 4463274232, 33886781840, 258260802232, 1974759985952, 15143163422794, 116417053435316, 896996316176290, 6925241271855296, 53562550587963052, 414948608904171464, 3219356873886333676

Offset: 0

Author

Paul D. Hanna, Sep 03 2012

Keywords

• nonn
• easy

Examples

			G.f.: A(x) = 1 + 2*x + 13*x^2 + 80*x^3 + 538*x^4 + 3740*x^5 + 26650*x^6 +...
where 1/A(x)^3 = 1 - 6*x - 15*x^2 - 8*x^3.
The logarithm of the g.f. begins:
log(A(x)) = 2*x + 22*x^2/2 + 170*x^3/3 + 1366*x^4/4 + 10922*x^5/5 + 87382*x^6/6 +...+ A007613(n)*x^n/n +...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A004987, A007613, A216317, A216357, A216358.

Programs

Mathematica

CoefficientList[Series[1/((1-8*x)*(1+x)^2)^(1/3), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

PARI

{a(n)=polcoeff(1/( (1-8*x)*(1+x)^2 +x*O(x^n) )^(1/3),n)}

PARI

{a(n)=local(A=1+x); A=exp(sum(m=1, n+1, sum(j=0, m, binomial(3*m, 3*j))*x^m/m +x*O(x^n))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} A007613(n)*x^n/n ), where A007613(n) = Sum_{k=0..n} binomial(3*n,3*k).

Recurrence: n*a(n) = (7*n-5)*a(n-1) + 8*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ Gamma(2/3)*2^(3*n+1)/(3^(5/6)*Pi*n^(2/3)). - Vaclav Kotesovec, Oct 20 2012

Inverse binomial transform of A004987. - Peter Bala, Jul 02 2023